Analysis of the descriptor Roesser model with the use of the Drazin inverse

Tadeusz Kaczorek

doi:10.1515/amcs-2015-0040

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Analysis of the descriptor Roesser model with the use of the Drazin inverse

International Journal of Applied Mathematics and Computer Science

Volume 25 (2015): Issue 3 (September 2015)

By: Tadeusz Kaczorek

Open Access

|Sep 2015

Abstract

A method of analysis for a class of descriptor 2D discrete-time linear systems described by the Roesser model with a regular pencil is proposed. The method is based on the transformation of the model to a special form with the use of elementary row and column operations and on the application of a Drazin inverse of matrices to handle the model. The method is illustrated with a numerical example

References

Bru, R., Coll, C., Romero-Vivo, S. and Sanchez, E. (2003). Some problems about structural properties of positive descriptor systems, in L. Benvenuti, A. de Santis and L. Farina (Eds.), Positive Systems, Lecture Notes in Control and Information Sciences, Vol. 294, Springer, Berlin, pp. 233-240.10.1007/978-3-540-44928-7_32
Search in Google Scholar
Bru, R., Coll, C. and Sanchez, E. (2000). About positively discrete-time singular systems, in M.E. Mastorakis (Ed.), System and Control: Theory and Applications, World Scientific and Engineering Society, Athens, pp. 44-48.
Search in Google Scholar
Bru, R., Coll, C. and Sanchez, E. (2002). Structural properties of positive linear time-invariant difference-algebraic equations, Linear Algebra and Its Applications 349: 1-10.10.1016/S0024-3795(02)00277-X
Search in Google Scholar
Busłowicz, M. (2014). Controllability, reachability and minimum energy control of fractional discrete-time linear systems with multiple delays in state, Bulletin of the Polish Academy of Sciences: Technical Sciences 62(2): 233-239.10.2478/bpasts-2014-0023
Search in Google Scholar
Campbell, S.L., Meyer, C.D. and Rose, N.J., (1976). Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients, SIAM Journal on Applied Mathematics 31(3): 411-425.10.1137/0131035
Search in Google Scholar
Dai, L. (1989). Singular Control Systems, Lectures Notes in Control and Information Sciences, Springer-Verlag, Berlin.10.1007/BFb0002475
Search in Google Scholar
Dodig, M. and Stosic, M. (2009). Singular systems state feedbacks problems, Linear Algebra and Its Applications 431(8): 1267-1292.10.1016/j.laa.2009.04.024
Search in Google Scholar
Duan, G.-R. (2010). Analysis and Design of Descriptor Linear Systems, Springer, New York, NY.10.1007/978-1-4419-6397-0
Search in Google Scholar
Fahmy, M.M, and O’Reill, J. (1989). Matrix pencil of closed-loop descriptor systems: Infinite-eigenvalues assignment, International Journal of Control 49(4): 1421-1431.10.1080/00207178908559713
Search in Google Scholar
Gantmacher, F.R. (1960). The Theory of Matrices, Chelsea Publishing Co., New York, NY.
Search in Google Scholar
Kaczorek, T. (1992). Linear Control Systems, Vol. 1, Research Studies Press J. Wiley, New York, NY.
Search in Google Scholar
Kaczorek, T. (2002). Positive 1D and 2D Systems, Springer-Verlag, London.10.1007/978-1-4471-0221-2
Search in Google Scholar
Kaczorek, T. (2004). Infinite eigenvalue assignment by an output feedback for singular systems, International Journal of Applied Mathematics and Computer Science 14(1): 19-23.
Search in Google Scholar
Kaczorek, T. (2010). Positive linear systems with different fractional orders, Bulletin of the Polish Academy of Sciences: Technical Sciences 58(3): 453-458.10.2478/v10175-010-0043-1
Search in Google Scholar
Kaczorek, T. (2011a). Checking of the positivity of descriptor linear systems by the use of the shuffle algorithm, Archives of Control Sciences 21(3): 287-298.10.2478/v10170-010-0044-1
Search in Google Scholar
Kaczorek, T. (2011b). Reduction and decomposition of singular fractional discrete-time linear systems, Acta Mechanica et Automatica 5(4): 62-66.
Search in Google Scholar
Kaczorek T. (2011c). Singular fractional discrete-time linear systems, Control and Cybernetics 40(3): 753-761.
Search in Google Scholar
Kaczorek, T. (2011d). Selected Problems of Fractional Systems Theory, Springer-Verlag, Berlin.10.1007/978-3-642-20502-6
Search in Google Scholar
Kaczorek, T. (2013). Application of the Drazin inverse to analysis of descriptor fractional discrete-time linear systems with regular pencils, International Journal of Applied Mathematics and Computer Science 23(1): 29-33, DOI: 10.2478/amcs-2013-0003.10.2478/amcs-2013-0003
Search in Google Scholar
Kaczorek, T. (2014a). Drazin inverse matrix method for fractional descriptor continuous-time linear systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 62(3): 409-412.10.2478/bpasts-2014-0042
Search in Google Scholar
Kaczorek, T. (2014b). Minimum energy control of descriptor positive discrete-time linear systems, COMPEL 33(3): 1-14.10.1108/COMPEL-04-2013-0111
Search in Google Scholar
Kaczorek, T. (2014c). Minimum energy control of positive fractional descriptor continuous-time linear systems, IET Control Theory and Applications 8(4): 219-225.10.1049/iet-cta.2013.0362
Search in Google Scholar
Kucera, V. and Zagalak, P. (1988). Fundamental theorem of state feedback for singular systems, Automatica 24(5): 653-658.10.1016/0005-1098(88)90112-4
Search in Google Scholar
Van Dooren, P. (1979). The computation of Kronecker’s canonical form of a singular pencil, Linear Algebra and Its Applications 27: 103-140.10.1016/0024-3795(79)90035-1
Search in Google Scholar
Virnik, E. (2008). Stability analysis of positive descriptor systems, Linear Algebra and Its Applications 429(10): 2640-2659.10.1016/j.laa.2008.03.002
Search in Google Scholar